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Properties

Label 2-2592-3.2-c0-0-1
Degree $2$
Conductor $2592$
Sign $-i$
Analytic cond. $1.29357$
Root an. cond. $1.13735$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.93i·5-s + 1.73·13-s + 0.517i·17-s − 2.73·25-s + 0.517i·29-s + 37-s − 1.41i·41-s − 49-s + 1.41i·53-s − 61-s + 3.34i·65-s − 1.73·73-s − 0.999·85-s + 1.93i·89-s − 1.41i·101-s + ⋯
L(s)  = 1  + 1.93i·5-s + 1.73·13-s + 0.517i·17-s − 2.73·25-s + 0.517i·29-s + 37-s − 1.41i·41-s − 49-s + 1.41i·53-s − 61-s + 3.34i·65-s − 1.73·73-s − 0.999·85-s + 1.93i·89-s − 1.41i·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2592\)    =    \(2^{5} \cdot 3^{4}\)
Sign: $-i$
Analytic conductor: \(1.29357\)
Root analytic conductor: \(1.13735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2592} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2592,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.249637285\)
\(L(\frac12)\) \(\approx\) \(1.249637285\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 1.93iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 1.73T + T^{2} \)
17 \( 1 - 0.517iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 0.517iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.73T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.93iT - T^{2} \)
97 \( 1 + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.283687659218333133334932221320, −8.419215886236822913275994690681, −7.62249591586453300543186867180, −6.92002666290696829676584282065, −6.18149841198059351207588477153, −5.77274647662331976537431601963, −4.23001315107718891240405761203, −3.49481691729189560328837598482, −2.81354725631869922620911950990, −1.65982862005967675453417260060, 0.893359783102715216522950490409, 1.76529301808682896648500888353, 3.30131844448192072836828121224, 4.28920629871198502152498334337, 4.83431023790833045164887381187, 5.78204119813552837580542365884, 6.31351530272855177733033964083, 7.63840617599287154503483993983, 8.288710628327956165867583659676, 8.796926749919001309642402211323

Graph of the $Z$-function along the critical line